Computational Arithmetic of Modular Forms - Mathematics Research

111 Pages·2008·0.81 MB·English

Computational Arithmetic of Modular Forms
(Modulformen II)
Wintersemester2007/2008
UniversitätDuisburg-Essen
GaborWiese
[email protected]
Versionof4thFebruary2008
2
Preface
Thislectureisaboutcomputingmodularformsandsomeoftheirarithmeticproperties.
Wesetthefollowingchallengingobjectives:
• We explain and completely prove the modular symbols algorithm in as elementary and as ex-
plicittermsaspossible. Thechosenapproachisbasedongroupcohomology.
• Thedevotedstudentshallbeenabledtoimplementthe(groupcohomological)modularsymbols
algorithm over any ring (such that a sufficient linear algebra theory is available in the chosen
computeralgebrasystem).
• WeintroducethetheoryofGaloisrepresentationsattachedtomodularformsinasexplicitterms
as possible. We explain some of its number theoretic significance and some computational
approaches.
• ThedevotedstudentshallbeenabledtocomputeimportantpropertiesofGaloisrepresentations
attachedtomodularformsexplicitly.
Accordingtotheseobjectivesthelectureconsistsoftwomainparts:
I. ComputingModularForms
II. ComputationalGaloisRepresentations
Due to the diversity of the audience, ranging from students up to PhD students intending to gen-
eralise the presented algorithms in different directions, and due to the dual aims, theoretic and algo-
rithmic,thelectureisconceivedinparallellayers. Notalllayersneedbefollowedbyallstudentsand
alllayerscanbereducedindividually. Thelayersarethefollowing:
• Theory. Roughly in 3 of the 4h per week the lecture will introduce theoretical results. All
students are expected to attend the lectures. The lectures will be accompanied by exercises
concerningthetheorypresented. Exercisescanbehandedinandwillbecorrected. Sometime
willbedevotedtodiscussingpossiblesolutions.
• Algorithmsandimplementations. Inalectureinthebeginning,programminginsomestandard
computer algebra systems is introduced. In some lectures during the term, algorithms and
possibly concrete implementations are presented. Much emphasis is laid on practical issues
and students will also be asked to find and implement algorithms. Possible solutions will be
discussed.
• Self-learnmodules. Forthedevotedstudenttogainamorecompletepictureofthetheorythan
canbepresentedduringthelecture,complementaryreadingissuggested.
3
The parallel layers will not necessarily be on a single subject all the time, as it is often necessary to
introducetheoryfirst. Thelectureisdividedupintostages,insteadofchapters,inordertoemphasize
thepossiblevarietyofsubjectsineachstage.
TheconceptionofthislectureisdifferentfromeverytreatmentIknow,inparticular,fromWilliam
Stein’s excellent book “Modular Forms: A Computational Approach” ([Stein]). Parts will, however,
besimilartonotes ofa seriesof 4lectures thatIgaveattheMSRIGraduateWorkshopinComputa-
tionalNumberTheory“ComputingWithModularForms”([MSRI]). Weemphasizethecentralroleof
Heckealgebrasandfocusontheuseofgroupcohomology,sinceontheonehanditcanbedescribedin
veryexplicitandelementarytermsandontheotherhandalreadyallowstheapplicationofthestrong
machineryofhomologicalalgebra. Weshallmentiongeometricapproachesonlyinpassing.
Organisational issues will be discussed with all participants and decided together in order to suit
everybody.
Contents
1 MotivationandSurvey 6
1.1 Theory: BriefreviewofmodularformsandHeckeoperators . . . . . . . . . . . . . 6
1.2 Theory: Themodularsymbolsformalism . . . . . . . . . . . . . . . . . . . . . . . 11
1.3 Theory: Themodularsymbolsalgorithm . . . . . . . . . . . . . . . . . . . . . . . . 17
1.4 Theory: Numbertheoreticapplications . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.5 Theory: Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.6 AlgorithmsandImplementations: MAGMA andSAGE . . . . . . . . . . . . . . . . . 25
1.7 AlgorithmsandImplementations: ModularsymbolsinMAGMA . . . . . . . . . . . 25
1.8 Computerexercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.9 Self-learnmodule: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2 Heckealgebras 29
2.1 Theory: Heckealgebrasandmodularformsoverrings . . . . . . . . . . . . . . . . 30
2.1.1 Somecommutativealgebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.1.2 CommutativealgebraofHeckealgebras . . . . . . . . . . . . . . . . . . . . 37
2.2 AlgorithmsandImplementations: LocalisationAlgorithms . . . . . . . . . . . . . . 38
2.2.1 Primaryspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.2.2 Algorithmforcomputingcommonprimaryspaces . . . . . . . . . . . . . . 40
2.2.3 Algorithmforcomputingidempotents . . . . . . . . . . . . . . . . . . . . . 41
2.3 AlgorithmsandImplementations: MoreofMAGMA . . . . . . . . . . . . . . . . . . 42
2.4 Theoreticalexercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.5 Computerexercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3 Homologicalalgebra 45
3.1 Theory: CategoriesandFunctors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2 Theory: ComplexesandCohomology . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.3 Theory: CohomologicalTechniques . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.4 Theory: GeneralitiesonGroupCohomology . . . . . . . . . . . . . . . . . . . . . . 56
3.5 Theoreticalexercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4
CONTENTS 5
4 CohomologyofPSL (Z) 61
2
4.1 Theory: PSL (Z)asafreeproduct . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2
4.2 Theory: Mayer-VietorisforPSL (Z) . . . . . . . . . . . . . . . . . . . . . . . . . 62
2
4.3 Theory: Parabolicgroupcohomology . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.4 Theory: Dimensioncomputations . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.5 Theoreticalexercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.6 Computerexercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5 ModularsymbolsandManinsymbols 70
5.1 Theory: Maninsymbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.2 Theory: Maninsymbolsandgroupcohomology . . . . . . . . . . . . . . . . . . . . 74
5.3 AlgorithmsandImplementations: ConversionbetweenManinandmodularsymbols . 74
5.4 Theoreticalexercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.5 Computerexercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6 Eichler-Shimura 77
6.1 Theory: Peterssonscalarproduct . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.2 Theory: TheEichler-Shimuramap . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.3 Theory: CupproductandPeterssonscalarproduct . . . . . . . . . . . . . . . . . . . 85
6.4 Theory: TheEichler-Shimuratheorem . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.5 Theoreticalexercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
7 Heckeoperators 92
7.1 Heckerings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
7.2 Heckeoperatorsonmodularforms . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
7.3 Heckeoperatorsongroupcohomology . . . . . . . . . . . . . . . . . . . . . . . . . 98
7.4 Theory: Eichler-Shimurarevisited . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
7.5 Theory: TransferofHeckeoperatorstoManinsymbols . . . . . . . . . . . . . . . . 102
7.6 Theoreticalexercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
7.7 Computerexercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
8 ImagesofGaloisRepresentations 106
Stage 1
Motivation and Survey
This section serves as an introduction to the topics that we are planning to cover this term. We will
briefly review the theory of modular forms and Hecke operators. Then we will define the modular
symbolsformalismandstatethetheorembyEichlerandShimuraestablishingalinkbetweenmodular
forms and modular symbols. This link is the central ingredient for the first part of the lecture, since
the modular symbols algorithm for the computation of modular forms is entirely based on it. In this
introduction,weshallalreadybeabletogiveabriefoutlineofthisalgorithm.
Inthesecondpartoftheintroduction,wewillstateandexplainthetheoremsbyShimura,Deligne
and Serre attaching a Galois representation to a Hecke eigenform. The modern number theoretic
significanceofmodularformsarisesfromthesetheorems(e.g.theroleofmodularformsintheproof
ofFermat’sLastTheorem). Wewillalsosketchwhichnumbertheoreticinformationcanbeobtained
fromcomputingmodularforms.
In the practically oriented part of the lecture, we shall introduce the computer algebra systems
MAGMAandSAGEandalsoshowhowtousethemodularformsandmodularsymbolspackagesthat
arealreadyprovidedbythesesystems.
1.1 Theory: Brief review of modular forms and Hecke operators
Congruencesubgroups
WefirstrecallthestandardcongruencesubgroupsofSL (Z). ByN weshallalwaysdenoteapositive
2
integer.
Considerthegrouphomomorphism
SL (Z) → SL (Z/NZ).
2 2
By Exercise 1 it is surjective. Its kernel is called Γ(N). The group SL (Z/NZ) acts naturally on
2
(Z/NZ)2 (bymultiplyingthematrixwithavector). Inparticular,themapSL (Z/NZ) → (Z/NZ)2
2
givenby a b 7→ a b (1) = (a)takesall(a) ∈ (Z/NZ)2asimagesuchthata,cgenerateZ/NZ
c d c d 0 c c
(cid:0) (cid:1) (cid:0) (cid:1)
(that’sduetothedeterminantbeing1). Wealsopointoutthattheimagecanandshouldbeviewedas
6
1.1. THEORY:BRIEFREVIEWOFMODULARFORMSANDHECKEOPERATORS 7
the set of elements in (Z/NZ)2 which are of precise (additive) order N. We consider the stabiliser
of (1). We define the group Γ (N) as the preimage of that stabiliser group in SL (Z). Explicitly,
0 1 2
thismeansthatΓ (N)consistsofthosematricesinSL (Z)whosereductionmoduloN isoftheform
1 2
(1 ∗).
0 1
The group SL (Z/NZ) also acts on P1(Z/NZ), the projective line over Z/NZ which one can
2
define as the tuples (a : c) with a,c ∈ Z/NZ such that ha,ci = Z/NZ modulo the equivalence
relationgivenbymultiplicationbyanelementof(Z/NZ)×. Theactionisthenaturalone(weshould
actually view (a : c) as a column vector, as above). The preimage in SL (Z) of the stabiliser group
2
of (1 : 0) is called Γ (N). Explicitly, it consists of those matrices in SL (Z) whose reduction is of
0 2
the form (∗ ∗). We also point out that the quotient of SL (Z/NZ) modulo the stabiliser of (1 : 0)
0 ∗ 2
correspondstothesetofcyclicsubgroupsofpreciseorderN inSL (Z/NZ). Theseobservationsare
2
atthebaseofdefininglevelstructuresforellipticcurves(see[MF]).
Itisclearthat
a b 7→a
Γ (N)/Γ (N) −“−c−d−”−−→ (Z/NZ)×
0 1
isagroupisomorphism. Wealsolet
χ : (Z/NZ)× → C×
denote a character, i.e. a group homomorphism. We shall extend χ to a map (Z/NZ) → C by
imposingχ(r) = 0if(r,N) 6= 1.
ByclassfieldtheoryorExercise2wehavetheisomorphism
Gal(Q(ζ )/Q) −F−r−ob−l−7→→l (Z/NZ)×
N
forallprimesl ∤ N. Byζ wedenoteanyprimitiveN-throotofunity. Weshall, thus, later onalso
N
consider χ as a character of Gal(Q(ζ )/Q). The name Dirichlet character (here of modulus N) is
N
commonusageforboth.
Modularforms
We now recall the definitions of modular forms. We denote by H the upper half plane, i.e. the set
{z ∈ C|Im(z) > 0}. ThesetofcuspsisbydefinitionP1(Q) = Q∪{∞}.
ForM = a b anintegermatrixwithnon-zerodeterminant,anintegerkandafunctionf : H →
c d
(cid:0) (cid:1)
C,weput
az+b det(M)k−1
(f| M)(z) = (f|M)(z) := f .
k cz+d (cz+d)k
(cid:0) (cid:1)
Fixintegersk andN ≥ 1. Afunction
f : H → C
givenbyaconvergentpowerseries(thea (f)arecomplexnumbers)
n
∞ ∞
f(z) = a (f)(e2πiz)n = a qn withq(z) = e2πiz
n n
X X
n=0 n=0
8 STAGE1. MOTIVATIONANDSURVEY
iscalledamodularformofweightk forΓ (N)if
1
(i) thefunction(f| a b )(z) = f(az+b)(cz+d)−k isaholomorphicfunction(stillfromHtoC)
k c d cz+d
forall a b ∈ SL(cid:0) (Z(cid:1))(thisconditioniscalledf isholomorphicatthecuspa/c),and
c d 2
(cid:0) (cid:1)
(ii) (f| a b )(z) = f(az+b)(cz+d)−k = f(z)forall a b ∈ Γ (N).
k c d cz+d c d 1
(cid:0) (cid:1) (cid:0) (cid:1)
WeusethenotationM (Γ (N); C). Ifwereplace(i)by
k 1
(i)’ the function (f| a b )(z) = f(az+b)(cz + d)−k is a holomorphic function and the limit
k c d cz+d
f(az+b)(cz+d)−(cid:0)k is0(cid:1)whenz tendstoi∞(weoftenjustwrite∞),
cz+d
thenf iscalledacuspform. Forthese,weintroducethenotationS (Γ (N); C).
k 1
Let us now suppose that we are given a Dirichlet character χ of modulus N as above. Then we
replace(ii)asfollows:
(ii)’ f(az+b)(cz+d)−k = χ(d)f(z)forall a b ∈ Γ (N).
cz+d c d 0
(cid:0) (cid:1)
Functions satisfying this condition are called modular forms (respectively, cusp forms if they satisfy
(i)’) of weight k, character χ and level N. The notation M (N,χ; C) (respectively, S (N,χ; C))
k k
willbeused.
All these are finite dimensional C-vector space and for k ≥ 2, there are dimension formulae,
which one can look up in [Stein]. We, however, point the reader to the fact that for k = 1 nearly
nothing about the dimension is known (except that it is smaller than the respective dimension for
k = 2;itisbelievedtobemuchsmaller,butonlyveryweakresultsareknowntodate).
Heckeoperators
AtthebaseofeverythingthatwewilldowithmodularformsaretheHeckeoperatorsandthediamond
operators. Oneshouldreallydefinethemconceptually,aswehavedonein[MF]. Hereisadefinition
byformulae.
IfaisanintegercoprimetoN,byExercise3wemayletσ beamatrixinΓ (N)suchthat
a 0
σ ≡ a−1 0 mod N. (1.1.1)
a 0 a
(cid:0) (cid:1)
We define the diamond operator hai (you see the diamond in the notation, with some phantasy)
bytheformula
haif = f| σ .
k a
If f ∈ M (N,χ; C), then we have by definition haif = χ(a)f. The diamond operators give a
k
group action of (Z/NZ)× on M (Γ (N); C) and on S (Γ (N); C), and the M (N,χ; C) and
k 1 k 1 k
S (N,χ; C)aretheχ-eigenspacesforthisaction.
k
Letlbeaprime. Welet
R := { 1 r |0 ≤ r ≤ l−1}∪{σ l 0 }, ifl ∤ N (1.1.2)
l 0 l l 0 1
(cid:0) (cid:1) (cid:0) (cid:1)
R := { 1 r |0 ≤ r ≤ l−1}, ifl | N (1.1.3)
l 0 l
(cid:0) (cid:1)
1.1. THEORY:BRIEFREVIEWOFMODULARFORMSANDHECKEOPERATORS 9
WeusethesesetstodefinetheHeckeoperatorT actingoff asaboveasfollows:
l
f| T = T f = f| δ.
k l l k
X
δ∈Rl
Lemma1.1.1 Suppose f ∈ M (N,χ; C). Recall that we have extended χ so that χ(l) = 0 if l
k
dividesN. Wehavetheformula
a (T f) = a (f)+lk−1χ(l)a (f).
n l ln n/l
Intheformula,a (f)istobereadas0ifldoesnotdividen.
n/l
Proof. Exercise4. 2
TheHeckeoperatorsforcompositencanbedefinedasfollows(weputT tobetheidentity):
1
• Tlr+1 = Tl ◦Tlr −lk−1hliTlr−1 forallprimeslandr ≥ 1,
• T = T ◦T forcoprimepositiveintegersu,v.
uv u v
Wederivetheveryimportantformula(validforeveryn)
a (T f) = a (f). (1.1.4)
1 n n
Itistheonlyformulathatwewillreallyneed.
FromtheaboveformulaeitisalsoevidentthattheHeckeoperatorscommuteamongoneanother.
By Exercise 5 eigenspaces for a collection of operators (i.e. each element of a given set of Hecke
operators acts by scalar multiplication) are respected by all Hecke operators. Hence, it makes sense
to consider modular forms which are eigenvectors for every Hecke operator. These are called Hecke
eigenforms,oroftenjusteigenforms. Suchaneigenformf iscallednormalisedifa (f) = 1.
1
Weshallconsidereigenformsinmoredetailinthefollowingstage.
Finally,letuspointouttheformula(forlprimeandl ≡ d mod N)
lk−1hdi = T2−T . (1.1.5)
l l2
Hence, the diamond operators can be expressed as Z-linear combinations of Hecke operators. Note
thatdivisibilityisnotroublesincewemaychoosel ,l ,bothcongruenttodmoduloN satisfyingan
1 2
equation1 = lk−1r+lk−1s.
1 2
Heckealgebrasandtheq-pairing
We now quickly introduce the concept of Hecke algebras. It will be treated in more detail in later
sections. Infact,whenweclaimtocomputemodularformswiththemodularsymbolsalgorithm,we
are really computing Hecke algebras. In the couple of lines to follow, we, however, show that the
Hecke algebra is the dual of modular forms, and hence all knowledge about modular forms can - in
principal-bederivedfromtheHeckealgebra.
10 STAGE1. MOTIVATIONANDSURVEY
Forthemoment,wedefinetheHeckealgebraofM (Γ (N); C)asthesub-C-algebrainsidethe
k 1
endomorphism ring of the C-vector space M (Γ (N); C) generated by all Hecke operators and all
k 1
diamondoperators. WemakesimilardefinitionsforS (Γ (N); C),M (N,χ; C)andS (N,χ; C).
k 1 k k
Letusintroducethenotations
T (M (Γ (N); C)),T (S (Γ (N); C)),T (M (N,χ; C))andT (S (N,χ; C)),
C k 1 C k 1 C k C k
respectively.
Wenowdefineabilinearpairing,whichIcallthe(complex)q-pairing,as
M (N,χ; C)×T (M (N,χ; C)) → C, (f,T) 7→ a (Tf)
k C k 1
(comparewithEquation1.1.4).
Lemma1.1.2 Suppose k ≥ 1. The complex q-pairing is perfect, as is the analogous pairing for
S (N,χ; C). Inparticular,
k
∼
M (N,χ; C) = Hom (T (M (N,χ; C)),C), f 7→ (T 7→ a (Tf))
k C C k 1
andsimilarlyforS (N,χ; C). ForS (N,χ; C),theinverseisgivenbyφ 7→ ∞ φ(T )qn.
k k n=1 n
P
Proof. Let us first recall that a pairing over a field is perfect if and only if it is non-degenerate.
Thatiswhatwearegoingtocheck. ItfollowsfromEquation1.1.4likethis. Ifforallnwehave0 =
a (T f) = a (f),thenf = 0(thisisimmediatelyclearforcuspforms;forgeneralmodularformsat
1 n n
the first place we can only conclude that f is a constant, but since k ≥ 1, non-zero constants are not
modularforms). Conversely,ifa (Tf) = 0forallf,thena (T(T f)) = a (T Tf) = a (Tf) = 0
1 1 n 1 n n
for all f and all n, whence Tf = 0 for all f. As the Hecke algebra is defined as a subring in the
endomorphismofM (N,χ; C) (resp.thecuspforms), wefindT = 0, provingthenon-degeneracy.
k
2
Theperfectnessoftheq-pairingisalsocalledtheexistenceofaq-expansionprinciple.
TheHeckealgebraisthelineardualofthespaceofmodularforms.
Lemma1.1.3 Letf inM (Γ (N); C)beanormalisedeigenform. Then
k 1
T f = a (f)f foralln ∈ N.
n n
Moreover,thenaturalmapfromtheabovedualitygivesabijection
{NormalisedeigenformsinM (Γ (N); C)} ↔ Hom (T (M (Γ (N); C)),C).
k 1 C−alg C k 1
Similarresultshold,ofcourse,alsointhepresenceofχ.
Proof. Exercise6. 2